Gröbner Basis

A Gröbner basis for an ideal I in a polynomial ring over a field (with respect to a chosen monomial order) is a finite generating set whose leading terms generate the ideal of leading terms; equivalently it yields a confluent polynomial reduction algorithm so that multivariate division by the basis gives canonical remainders and decides ideal membership and elimination tasks.

A finite generating set of a polynomial ideal in a polynomial ring, chosen so that with respect to a fixed monomial order every polynomial in the ideal has a unique normal form modulo that set; in practice a Gröbner basis makes ideal membership, elimination, and computation of algebraic invariants algorithmic.